Abstract
The extraction of traveling wave solutions for nonlinear evolution equations is a challenge in various mathematics, physics, and engineering disciplines. This article intends to analyze several traveling wave solutions for the modified regularized long-wave (MRLW) equation using several approaches, namely, the generalized algebraic method, the Jacobian elliptic functions technique, and the improved Q-expansion strategy. We successfully obtain analytical solutions consisting of rational, trigonometric, and hyperbolic structures. The adaptive moving mesh technique is applied to approximate the numerical solution of the proposed equation. The adaptive moving mesh method evenly distributes the points on the high error areas. This method perfectly and strongly reduces the error. We compare the constructed exact and numerical results to ensure the reliability and validity of the methods used. To better understand the considered equation's physical meaning, we present some 2D and 3D figures. The exact and numerical approaches are efficient, powerful, and versatile for establishing novel bright, dark, bell-kink-type, and periodic traveling wave solutions for nonlinear PDEs.